Toric forms of elliptic curves and their arithmetic

Elliptic Curves and Algebraic Topology

Elliptic curves enter algebraic topology through “Elliptic cohomology”–really a family of cohomology theories–and their associated “elliptic genera”. • Arithmetic aspect: Modularity of elliptic genera, The spectrum TMF of “topological modular forms” and the calculation of π∗TMF →MF (Z), Hopkins’s proof of Borcherds’ congruences. • Physical aspect: Witten’s approach to elliptic genera via string...

An Efficient Threshold Verifiable Multi-Secret Sharing Scheme Using Generalized Jacobian of Elliptic Curves

‎In a (t,n)-threshold secret sharing scheme‎, ‎a secret s is distributed among n participants such that any group of t or more participants can reconstruct the secret together‎, ‎but no group of fewer than t participants can do‎. In this paper, we propose a verifiable (t,n)-threshold multi-secret sharing scheme based on Shao and Cao‎, ‎and the intractability of the elliptic curve discrete logar...

Efficient elliptic curve cryptosystems

Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...

Study of Finite Field over Elliptic Curve: Arithmetic Means

Public key cryptography systems are based on sound mathematical foundations that are designed to make the problem hard for an intruder to break into the system. Number theory and algebraic geometry, namely the theory of elliptic curves defined over finite fields, has found applications in cryptology. The basic reason for this is that elliptic curves over finite fields provide an inexhaustible s...

Elliptic Curves , Arithmetic Geometry , and Modular Forms